This tag is for those who are trying to prove or derive reduction formulas of integrals. Reduction formulas are often useful to those trying to integrate trigonometric, exponential, or rational functions raised to certain powers, or functions containing multiple variables.
A reduction formula is a way to reduce an integral involving a parameter, usually a non-negative integer $n$, to a similar with a smaller parameter, until the 'base' case is simple enough to be evaluated directly. Integration by parts is often very helpful.
Examples include:
The gamma function $\Gamma(n)=\displaystyle\int_0^{\infty} e^{-t} t^{n-1} dt$. Using integration by parts one has $\Gamma(n+1)=n\Gamma(n)$, with $\Gamma(1)=1$.
Let $I_n=\displaystyle\int \sin^n(x) dx$; use integration by parts with $u=\sin^{n-1}(x)$ to obtain $I_n = -\dfrac{\sin^{n-1}(x)\cos(x)}{n}+\left(\dfrac{n-1}{n}\right) I_{n-2}$, with the base cases $I_0 =x$ and $I_1 =-\cos(x)$ discarding constants.
One solution of the Basel problem, due to Euler, required evaluation of $$A_n = \displaystyle\int _0^1 \dfrac{x^{2n+1}}{\sqrt{1-x^2}}\, dx.$$ It was shown that $A_0=1$ and $A_{n}= \dfrac{2n}{2n+1}A_{n-1}$, whence a closed-form is $A_n= \dfrac{2^n n!}{(2n+1)!!}.$
